On the second day of the year, Danielle Miller gave up on the federal government.

Furloughed from her Internal Revenue Service job near Cincinnati and fearful of running out of money during the partial government shutdown, she filed for unemployment benefits: $414 a week, about $200 less than usual.

“Once Christmas came and went, after New Year’s, I was like, I can’t go on,” said Ms. Miller, a single mother who has worked for the I.R.S. for almost 14 years. She spent part of this week calculating when her first unemployment check would arrive. “It’s disappointing, and it’s frustrating,” she said. “I have a job.”

The shutdown, the longest on record, is prompting tens of thousands of federal employees to seek jobless benefits. As the impasse meanders through its fourth week and more bills come due, their numbers have been growing.

“We’re certainly prepared for an influx of claims,” said William Walton, the director of unemployment insurance services at the Virginia Employment Commission. “The logical conclusion is that the longer the partial federal shutdown continues, the more claims we’re going to see.”

The trend is especially acute in the District of Columbia and neighboring Maryland and Virginia, which have many federal offices and installations. But a survey of state labor agencies by The New York Times shows that the shutdown is driving a mounting wave of unemployment filings across the country.

Texas has received more than 2,900 claims from federal workers since the shutdown began on Dec. 22, while Ohio is approaching 700. Kansas reported 445 filings, and Alabama was closing in on 500. Montana said it had logged almost 1,500.

But those may only be the beginning. About 380,000 furloughed federal workers are eligible for benefits, and if the White House and Congress remain at odds over President Trump’s demand for a wall along the southwestern border, state officials expect many thousands more applications.

“We need to be cognizant and responsive to our customers’ needs,” said Jeff Fitzgerald, who oversees the unemployment insurance program in Colorado, which has so far received about 2,200 federal claims. “At present, there is no foreseeable end to the customers’ needs, so, yeah, we’re bracing for that.”

Federal workers and the leaders of their unions said this week that people had begun to shift their thinking about how to get through the shutdown, which has not had a swift resolution. They said they were increasingly giving up on the idea that short-term stopgaps like borrowing from family and friends or postponing purchases would be enough.

“This week they’re projecting, ‘What do I need to do this week to pay February’s mortgage and February’s utilities?’” said Steve Reaves, a Federal Emergency Management Agency employee who leads the union for FEMA workers.

Mr. Reaves, who lives in Texas, prepared his own unemployment claim over the weekend.

“We need help on groceries, we need help on utility bills,” said Mr. Reaves, who said he did not apply for unemployment during the 16-day shutdown in 2013. “We’re cutting back on everything,” he said.

Federal employees are covered by a separate unemployment insurance system parallel to that for other workers. But the two systems are nearly identical, and are administered similarly by the states. A significant difference is that federal agencies, rather than private employers, bear the costs of the federal system.

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An agent at Colorado’s Department of Labor and Employment fielded calls in Denver. The agency has set up a team of nine employees just to handle inquiries from federal workers.CreditNick Cote for The New York Times

The Labor Department and many employment lawyers say that filing a claim now is not an option for all of the roughly 420,000 federal employees who are “essential” and are working without pay during the partial shutdown. According to the department, workers in those circumstances are “generally not eligible” to receive benefits.

There were not yet signs that many federal workers were rushing to drain retirement accounts.

The Thrift Savings Plan, a government retirement program with about 5.5 million participants, has received more than 5,500 requests for hardship withdrawals since Dec. 26, but Kim Weaver, a spokeswoman for the plan, said that was only a slight increase over the same period a year ago, and could not yet be directly tied to the shutdown. Ms. Weaver noted that the plan’s enrollment had grown since last year, and that in response to recent natural disasters, the I.R.S. had made hardship withdrawals easier to obtain.

In state unemployment offices, administrators said they expected the cost of the shutdown to mount every day. They said that earlier shutdowns like the one in 2013 and the plentiful near-misses over the years had taught them how to prepare for this one.

“What we have, unfortunately, working for us is experience,” said Cher Roybal Haavind, a spokeswoman for the Colorado Department of Labor and Employment, which set up a nine-person team in Denver to answer inquiries from federal workers.

Some state officials said they had asked utilities and other companies to extend mercy to federal employees, and the federal Office of Personnel Management — essentially the government’s human resources department — published sample letters that furloughed employees could send to creditors to ask for patience.

Still, officials said unemployment insurance usually offers the surest way to assist furloughed workers, even though they can be reluctant to apply. It can take weeks for benefits to start flowing, and then, when the shutdown ends and retroactive pay is granted, workers will generally have to reimburse the state for their unemployment benefits.

“They’re appreciative of our support and our service, but they’re very frustrated they even have to do it,” said Suzan G. LeVine, the employment security commissioner in Washington State, where about 2,200 federal workers have applied for benefits. “They have a job — why should they have to get unemployment insurance?”

Ms. LeVine said that in her state it typically takes two to three weeks for a claim to be processed. Although she and officials from other states said they were trying to expedite the work, the shutdown came at a busy time for them, when they were already coping with high seasonal volumes of claims from agriculture and construction workers.

Governors, including Gavin Newsom of California, where there are about 245,000 federal employees, have been urging workers to file claims; so have union leaders like Mr. Reaves, who worry about backlogs and waiting periods.

“What I’ve been preaching to them is to get it, because it will make ends meet until the furlough is over,” Mr. Reaves said of his members. He said he knew of a handful of FEMA employees, though, who decided not to wait out the shutdown and found other jobs.

Ms. Miller, the furloughed I.R.S. worker, said she hoped that unemployment benefits would help her avoid a similar reckoning. Two of her three children live with her in Covington, Ky., just across the Ohio River from Cincinnati, and she said she had already borrowed money from her parents to pay for groceries while waiting for her benefit checks to start.

“I would like to think it would not go on for weeks or months, but at this point, I have no idea,” she said of the shutdown. “I’m playing it day by day. I don’t want to have to find another job. I’ve put in 14 years at this place.”

When she went to apply for unemployment online, she said, the state website asked why she needed help.

“There was no option for shutdown,” she said. “It’s frustrating because it’s not that I don’t want to work, and it’s not that I can’t work. It’s that I’m not allowed to work.”

A version of this article appears in print on , Section A, Page 16 of the New York edition with the headline: With Little Left to Cut Back On, Federal Workers File for Unemployment. Order Reprints | Today’s Paper | Subscribe